a general mathematical model for analyzing the performance

8/8/2019 A General Mathematical Model for Analyzing the Performance

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A general mathematical model for analyzing the performanceof fuel-cell membrane-electrode assemblies

Huayang Zhu*, Robert J. KeeEngineering Division, Colorado School of Mines, Golden, CO 80401, USA

Received 9 January 2003; accepted 27 January 2003

Abstract

We have developed a general mathematical model to represent the membrane-electrode assembly (MEA) of fuel-cell systems. The model is

used to analyze the effects of various polarization resistances on cell performance. The model accommodates arbitrary gas mixtures on theanode and cathode sides of the MEA. Moreover, it accommodates a variety of porous electrode and electrolyte structures. Concentration

overpotentials are based on a dusty-gas representation of transport through porous electrodes. The activation overpotentials are represented

using the ButlerVolmer equation. Although the model is general, the emphasis in thispaper is on solid-oxide fuel-cell (SOFC) systems for the

direct electrochemical oxidation (DECO) of hydrocarbons.

# 2003 Elsevier Science B.V. All rights reserved.

Keywords: Model; MEA; Fuel cell; SOFC

1. Introduction

Quantitative models can be valuable in the interpretation

of experimental observations and in the development andoptimization of systems. The objective of the membrane-

electrode assembly (MEA) model developed herein is to

provide a physically based model that can be used to

evaluate the effects of design and operating alternatives.

While the model can stand alone, the software is written in a

way that enables incorporation into larger system-level

models that also consider fluid flow and thermal manage-

ment throughout a fuel-cell system. The model is general in

the sense that it can accommodate a variety of geometrical

configurations. It can also represent proton-conducting or

oxygen-ion-conducting electrolytes. The discussion and

examples in this paper, however, consider solid-oxide

fuel-cell (SOFC) system.

The great advantage of SOFC systems for highly efficient

electric power generation lies in its potential for direct use of

hydrocarbon fuels, without the requirement for upstream

fuel preparation, such as reforming [14]. A typical layout

for a planar design of the SOFC system is illustrated in Fig. 1.

The membrane-electrode assembly, which is composed of

an electrolyte sandwiched between the anode and the cath-

ode, is itself sandwiched between metal interconnect struc-

tures. The fuel and air channels are formed in the

interconnect structure. The electrolyte is a dense ceramic

(e.g. yttria-stabilized zirconia (YSZ) or gadolinia-doped

ceria (GDC)), which is impermeable to gas flow but is anoxygen-ion (O2) conductor. The composite electrodes are

porous metal-loaded ceramics (cermets) (e.g. NiYSZ for

the anode and Sr-doped LaMnO3 (LSM) for the cathode),

which may be mixed electronic and ionic conductors that

can extend the three-phase boundaries (TPB) and promote

the electrocatalytic reactions.

The SOFC system involves complex transport, chemical,

and electrochemical processes, with its operating perfor-

mance strongly affected by the corresponding transport

resistances and the activation barriers. Fig. 2 illustrates some

of these processes at the MEA level. During operation,

oxygen from the air channel transports through the porous

cathode, whereupon it is reduced to oxygen ion (O2) at

gascathodeelectrolyte three-phase boundaries. The

formed oxygen ion at the cathodeelectrolyte interface is

then transported to the anodeelectrolyte interface through

the ion-conducting electrolyte. At the same time, fuel from

the fuel channel transports through the porous anode to the

anodeelectrolyte interface, where it is oxidized electroche-

mically at the gasanodeelectrolyte three-phase boundaries

to products. The products (e.g. H2O and CO2) are then

transported back to the fuel channel through the porous

anode structure. Transport resistances of the gas-phase

species in the porous electrodes and O2 in the electrolyte,

Journal of Power Sources 117 (2003) 6174

* Corresponding author. Tel.: 1-303-273-3890; fax: 1-303-273-3602.E-mail address: [email protected] (H. Zhu).

0378-7753/03/$ see front matter # 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0378-7753(03)00358-6


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as well as the activation energy barriers for electrochemical

reactions can result in various polarizations. They are repre-

sented as concentration overpotentials, activation overpo-

tentials, and ohmic overpotentials. The main contribution to

the ohmic polarization is from the transport resistance of

O2 in the electrolyte. The concentration polarization is due

to the transport resistance of the gaseous species through

porous composite electrodes, and the activation polarization

is related to the charge-transfer processes at the electrode

electrolyte interfaces. These various polarizations are func-

tions of both operating conditions and the physical proper-

ties of the cell. The operating conditions include

temperature, pressure, and fuel and oxidizer concentrations.

The cell properties involve materials, macro- and micro-structures of the electrolyte and the composite electrodes,

which include porosity, tortuosity, permeability, and thick-

ness of the anode and the cathode, the ionic conductivity and

thickness of the electrolyte, the active area and activity of the

electrodeelectrolyte interface.

To understand these various resistances and enhance the

cell performance, parametric studies for these various polar-

izations in the electrolyte and composite electrodes have

been widely made through experiments [5,6], theoretical

analysis [7], and numerical models [810]. The theoretical

analyses and numerical models for the activation polariza-

tions require an understanding the elementary thermal

electrochemical reaction mechanisms and the microstruc-

ture of the electrode. Due to the complex and uncertain

details of the reaction processes (particularly in case of

hydrocarbons), most modeling in the literature has focused

on using H2 as the fuel. Chan et al. [8] used the intrinsic

charge-transfer resistance to represent the activation polar-

ization. Tanner et al. [11] have derived an effective charge-

transfer resistance for the composite electrode, which is a

function of the microstructural parameters of the electrode,

intrinsic charge-transfer resistance, ionic conductivity of theelectrolyte, and the electrode thickness. This effective

charge-transfer resistance model was later used to analyze

the activation overpotentials [9,10]. Adler et al. [12] derived

a charge-transfer resistance model for the composite cath-

ode, which is a function of the tortuosity, porosity, interface

area, oxygen self-diffusion coefficient and oxygen exchange

coefficient. The activation polarizations for a particular

electrode may also be measured by the electrochemical

impedance spectroscopy (EIS) [5,6,13]. Previous porous-

media gas-phase transport models have been developed to

predict concentration overpotentials [810]. These include

molecular diffusion, Knudsen diffusion, and pressure-driven

Darcy flow. However, these have only been applied in fuel-

cell systems with binary gas-phase mixtures.

The model developed in this paper represents a unit-cell

structure (Fig. 2), considering the effects of various over-

potentials on the cell performance. As illustrated in Fig. 2

the cell is anode-supported, with a very thin electrolyte and

cathode, and both electrodes are porous ceramicmetal

composite structures. Channels formed in an interconnect

structure carry the flow of fuel and air. Depending on the

position of the unit cell within a fuel-cell system, the

composition in both the fuel and air channels varies depend-

ing on initial composition and depletion.

Fig. 1. Typical layout of a planar SOFC. The left-hand panel illustrates the system at the scale of flow channels (in mm), while the right-hand panel illustrates

the small-scale structure (in mm) of the porous cermet electrode.

Fig. 2. Illustration of the components of the MEA unit cell.

62 H. Zhu, R.J. Kee / Journal of Power Sources 117 (2003) 6174


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2. Model formulation

The overall performance of SOFC systems is greatly

affected by ohmic, concentration, and activation polariza-

tions. Thus, the operating cell potential (Ecell) is lower than

the Nernst potential (E). The operating cell potential (Ecell)

can be formally expressed as:

Ecell E Zconc;a Zact;a Zohm Zconc;c Zact;c

Zinterface Zleakage; (1)

where Zconc;a and Zconc;c are the concentration overpotentials

at the anode and the cathode due to the gas-phase species

diffusion resistance, Zact;a and Zact;c the corresponding acti-

vation overpotentials, Zohm the ohmic overpotential in the

electrolyte, Zinterface the interface overpotential due to the

contact resistance at material boundaries, andZleakage the cell

potential loss due to the electric current through the electro-

lyte. Subsequent sections in the paper develop models for

each of these overpotentials, all of which are functions of thecurrent density ie.

2.1. Global electrochemical reaction

A global electrochemical reaction can be written as:

nf0XKk1

nf;kwk

! no

0XKk1

no;kwk

!?

XKk1

n00f;kwk XKk1

n00o;kwk;

(2)

where wk is the chemical symbol for the kth species (which

may participate as a fuel, an oxidizer, or both),nf0 and no

0 the

stoichiometric coefficients for the fuel and oxidizer mix-tures, and n00f;k and n

00o;k the stoichiometric coefficients of

the kth product species in the fuel and air channels, res-

pectively. Each species wk has a primary identity as a fuel,

an oxidizer, a product, or an inert, and this identity plays

an essential role in balancing the reaction to determine

stoichiometric coefficients and in assigning charge trans-

fer. The global reaction can be expressed in compact form

as:

XK

k1

nk0wk?

XK

k1

n00kwk; (3)

where nk0 nf

0nf;k no0no;k and n

00k n

00f;k n

00o;k.

Fuel and oxidizer mixtures are separated by the anode

electrolytecathode MEA structure. Thus, in writing a glo-

bal electrochemical reaction and calculating the Nernst

potential, it is convenient to represent the fuel mixture

and the oxidizer mixture separately as reactants. This

separation is accomplished by writing the LHS of the

reaction as two summations. The fuel channel mixture

composition is represented in terms of the species mole

numbers as nf;k. Similarly, no;k is used to represent the

mixture of the composition of the oxidizer mixture in the

air channel (e.g. for air, no;O2 0:21 and no;N2 0:79).

The RHS of the global reaction is also separated into two

terms. Depending on whether the cell is an oxygen-ion

conductor or a proton conductor, the product species appear

on the anode or cathode side of the MEA. Thus, following

reaction balancing, the location of the product species

affects the product stoichiometric coefficients n00f;k and

n00o;k. There is more discussion on this point following the

discussion on reaction balancing.

Since the electrochemical reactions involve charge trans-

fer it is necessary to specify the charge transfer associated

with each reactant species. For this purpose, we introduce

two sets of parameters (zf;k and zo;k), which specify the

charge transferred per mole of the kth fuel and oxidizer

species. For a species that is inert relative to the electro-

chemical charge-transfer process, z 0. In our convention,the product species (e.g. H2O and CO2) also are assigned

z 0. Consider two examples for assigning z. Since O2 inthe air channel is reduced to O2 by obtaining four electrons

(i.e. O2 4e ! 2O2), the parameter zo;O2 4. Because

the direct electrochemical reaction of CH4 with O2 at the

anode to produce H2O and CO2 (i.e. CH4 4O2 !

CO2 2H2O 8e) releases eight electrons, zf;CH4 8.

The total number of electrons transferred by the global

electrochemical reaction is represented as:

ne XKk1

nf0nf;kzf;k

XKk1

no0no;kzo;k: (4)

There may be reasons to incorporate oxidizer species into

the fuel channel or fuel species into the oxidizer channel. For

example, oxygen in the fuel channel may promote beneficial

partial oxidation of the fuel or it may work to inhibit depositformation. Such species, while influencing thermal chem-

istry in the gas or on catalytic surfaces, do not participate

directly in the electrochemical reaction. Thus, they are

considered inert in the global electrochemical reaction.

For example, if O2 (i.e. an oxidizer species) appears as part

of the fuel composition (i.e. in the first term of the global

reaction (2)), then zf;O2 0. The added oxygen in the fuelchannel cannot work directly to produce electric energy

only fuel oxidization with oxidizer from the cathode channel

can produce power. Thus, any oxidizer in the fuel channel

must reduce the cell potential because of its diluting effect.

Before proceeding to evaluate the cell potential, the global

reaction must be balanced to determine the stoichiometric

coefficients. Although balancing reactions is a well known

process, a few words of explanation are warranted. A

balance equation is written for each element (e.g. H, C,

O, N), requiring that the reaction conserves the element

balance. The process leads to a system of linear equations for

the stoichiometric coefficients (nf0, no

0, and n00k). Because of

an essential linear dependence stemming from the fact that

species are composed of elements in a specified way, one can

arbitrarily set nf0 1. Then depending on the number of

product species, only some of the n00k are non-trivial. Spe-

cifically, the number of product species (and hence the

H. Zhu, R.J. Kee / Journal of Power Sources 117 (2003) 6174 63


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number n00k coefficients) is one fewer than the number of

elements. Once the overall product stoichiometric coeffi-

cients n00k are known, then either:

n00f;k n00k or n

00o;k n

00k; (5)

depending on the type of cell (i.e. oxygen ion or proton

conducting) and hence the location of the product species.Inert species (i.e. reactantspecies for which zf;k 0 and

zo;k 0) are initially excluded from the balancing proce-dure. Once the balance is completed, the inert species are

reintroduced with the same stoichiometric coefficients on

the reactant and product sides of the global reaction. The

entire balancing process is completely automated in the

software.

2.2. Nernst potential and the Nernst equation

Based on the chemical potential balance at open circuit,

the Nernst equation provides that:

E E RT

neF

XKk1

nf0nf;k no

0no;k n00k ln

pk

p0

; (6)

where E is the Nernst electric potential, E the ideal Nernst

potential at standard conditions (p0 1 atm), pk the partialpressure of the kth species, T the temperature, R 8:314 J/(mol K) is the universal gas constant, and F 96485:309 C/mol is Faradays constant. The ideal Nernst potential at the

standard conditions is given as:

E DG

neF; (7)

where DG is the change in standard-state Gibbs free energy

between products and reactants of the global reaction Eq. (3).

Specifically:

DG XKk1

nf0nf;k no

0no;k n00km

k; (8)

where mk is the standard-state chemical potential of the kth

species. The standard-state thermodynamic properties of

ideal gases depend on temperature. The needed thermody-

namic properties are readily available in databases such as in

CHEMKIN [14]. As an alternative to using the species partial

pressures, the Nernst equation can also be represented interms of the species molar concentrations Xk as:

E E RT

neF

XKk1

nf0nf;k no

0no;k n00k lnXk

RT

neFln

RT

p0

XKk1

nf0nf;k no

0no;k n00k: (9)

Fig. 3 illustrates the Nernst potentials for three fuels as a

function of fuel utilization (at T 750 8C and p 1 atm).Consider the situation for butane where the global reaction is:

C4H10 132

O2 ! 5H2O 4CO2: (10)

If the fuel is 50% utilized then the fuel mixture is composed

of 1/2 moles of C4H10, 5/2 moles of H2O, and 4/2 moles of

CO2. The Nernst potentials are highest for the hydrocarbons,

assuming direct electrochemical oxidation. The figure also

shows that the Nernst potential for hydrogen is reduced most

as a function of fuel depletion. The potentials illustrated in

Fig. 3 are calculated presuming that the oxidizer is pure air

(i.e. no depletion of the oxygen in the air).

2.3. Concentration polarization

For open-circuit conditions (i.e. zero current flow), thespecies concentrations at the electrolyte interface (three-

phase boundary) are the same as those in the bulk channel

flow. Denoting the molar species concentrations in the

channels as Xk

, the Nernst potential is written as:

E E RT

neF

XKk1

nf0nf;k no

0no;k n00k lnXk

RT

neFln

RT

p0

XKk1

nf0nf;k no

0no;k n00k: (11)

However, when the current is flowing there must be con-

centration gradients across the electrode structures. This isbecause the diffusion processes that supply the species flux

to the electrochemical reactions are driven by concentration

gradients. Consequently, species concentrations at the three-

phase boundaries Xks

are different from the bulk concen-

trations. Evaluated at the three-phase boundaries, the corre-

sponding Nernst equation is written as:

Es E RT

neF

XKk1

nf0nf;k no

0no;k n00k lnXk

s

RT

neFln

RT

P0

X

K

k1

nf0nf;k no

0no;k n00k: (12)

Fig. 3. Nernst potential for three fuels-air systems as a function of

percentage of the fuel utilization. As the fuel is utilized it is converted to

stoichiometric products that dilute the fuel on the anode side. The air is not

depleted in this system.



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The potential difference associated with the concentration

variation is written in terms of a concentration overpotential

as follows:

Zconc E Es

RT

neFX

K

k1

nf0nf;k no

0no;k n00

f;k n00

o;k ln

Xk

Xks :

(13)

The equation can easily be rewritten to separate the anode

and cathode contributions as:

Zconc RT

neF

XKk1

nf0nf;k n

00f;k ln

Xk

Xks

RT

neF

XKk1

no0no;k n

00o;k ln

Xk

Xks

: (14)

Furthermore, the concentration overpotentials at the anode

and the cathode can be defined separately as:

Zconc;a RT

neF

XKk1

nf0nf;k n

00f;k ln

Xk

Xks

; (15)

Zconc;c RT

neF

XKk1

no0no;k n

00o;k ln

Xk

Xks

: (16)

There are a number of physical processes that may con-

tribute to the concentration polarization. These include gas

species molecular transport in the electrode pores, solution

of reactants into the electrolyte and dissolution of products

out of the electrolyte, and the diffusion of the reactants/

products through the electrolyte to/from the electrochemicalreaction sites. Typically, however, the diffusive transport of

the reactants and products between the flow channels and the

electrochemical reaction sites across the electrode is the

major contributor to the concentration polarization. For an

anode-supported MEA structure, the concentration polari-

zation at the cathode is usually very small due to its being

very thin (e.g. 50 mm). However, the concentration polar-

ization at the anode can be very significant, particularly at

the high current density and high fuel utilization rate.

2.3.1. Dusty-gas model

The concentration polarization is caused primarily by the

transport of gaseous species through porous electrodes. It is

affected by the microstructure of the electrodes, particularly,

the porosity, permeability, pore size, and tortuosity factor.

There are four major physical processes that affect transport

in the porous media. They are molecular diffusion, Knudsen

diffusion, surface diffusion, and Darcys viscous flow. The

relative importance of bulk diffusion (e.g. Fickian) and

Knudsen diffusion depends on the relative frequency of

molecular-molecular collisions and molecular-wall colli-

sions. These processes can be characterized by the Knudsen

number Kn l=d, where l is the mean free path length inthe gas and d is a characteristic pore diameter. When

Kn ( 1 bulk diffusion is dominant and when Kn ) 1Knudsen diffusion is dominant. When Kn % 1, which isthe typical situation in SOFC electrodes, both bulk diffusion

and Knudsen diffusion are comparable and must be con-

sidered together. The dusty-gas model (DGM) [15] was

developed to represent multicomponent transport in this

situation.The dusty-gas model can be used to describe the relation-

ship between the gas species concentrations in the flow

channel and concentrations at the electrochemical reaction

sites (at the electrodeelectrolyte interface). The DGM was

developed initially for astrophysical applications, assuming

that the pore wall consists of giant, motionless, and uni-

formly distributed particles (i.e. dust) in space. Three gas-

phase transport mechanisms are considered: molecular dif-

fusion, Knudsen diffusion, and viscous flow. Bulk diffusion

and Knudsen diffusion are combined in series to form the

total diffusive flux. The viscous porous-media flow (Darcy

flow) acts in parallel with the diffusive flux. There are three

porous-media parameters that appear in the DGM. They are

the Knudsen coefficient (Kg), the permeability constant (Bg),

and the porosity-to-tortuosity ratio (fg=tg), all which can bedetermined experimentally. The DGM can be written as an

implicit relationship among the molar concentrations, molar

fluxes, concentrations gradients, and the pressure gradient

[16,17]:

X6k

XNg

k XkNg

XTDek

Ng

k

Dek;Kn

rXk Xk

De

k;Kn

Bg

mrp: (17)

In this relationship Ng

k is the molar flux of species k, Xk themolar concentrations, and XT p=RT is the total molarconcentration. The mixture viscosity is given as m (which

depends on the mixture composition and temperature) and

Dek and Dek;Kn are the effective molecular binary diffusion

coefficients and Knudsen diffusion coefficients.

The effective molecular binary diffusion coefficients in

the porous media Dek are related to the ordinary binary

diffusion coefficients Dk in the bulk phase [18] as:

Dek fg

tg

Dk: (18)

Binary diffusion coefficients, which are determined from

kinetic theory, may be evaluated with software such as

CHEMKIN [18]. Knudsen diffusion, which occurs due to

gaswall collisions, becomes dominant when the mean free

path of the molecular species is much larger than the pore

diameter. The effective Knudsen diffusion coefficient can be

expressed as:

Dek;Kn 4

3Kg

ffiffiffiffiffiffiffiffiffi8RT

pWk

r; (19)

where the Knudsen permeability coefficient Kg rpfg=tg



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(fg is the porosity and tg the tortuosity factor), Wk are the

molecular weights, and rp the average pore radius.

Assuming that the porous electrode is formed by closely

packed spherical particles with diameter dp (an idealization),

the permeability can be expressed by the KozenyCarman

relationship as [19]:

Bg f3gd

2p

72tg1 fg2: (20)

Other porous-media situations have different permeabilities.

An iterative procedure is needed to determine the pressure

and concentration gradients. Assuming that the species

molar fluxes Ngk through the porous electrode are known

based on the current density and the global electrochemical

reaction at the electrodeelectrolyte interface, the pressure

gradient can be determined as:

rp PkN

g

k=Dek;Kn

1=RT Bg=mP

kXk=De

k;Kn

: (21)

This expression for the pressure gradient comes from a

summation of Eq. (17) over all species k, recognizing that

summation of the first term (representing ordinary diffu-

sion) vanishes exactly. The concentrations Xk in thedenominator are evaluated either in the channel or at the

electrolyte interface, depending on the direction of the net

molar fluxP

k Ng

k. If the net flux is toward the electrolyte (as

it normally is in the cathode), then the Xk are taken as thechannel concentrations. However, when the flux is toward

the channel (as it usually is in the anode), the concentrations

are evaluated at the electrolyte interface. The iteration is

needed because the electrolyte interface concentrations arenot yet known. Assuming an initial guess for the concen-

trations Xk (usually the channel concentrations), the pres-sure gradient is evaluated. Then, using this pressure

gradient, the concentration gradients can be determined

from Eq. (17) as:

rXk X6k

XNg

k XkNg

XTDek

Ng

k

Dek;Kn

Xk

Dek;Kn

Bg

mrp: (22)

The concentrations in the summation on the RHS are

evaluated as discussed above depending on the direction

of the net flux. This analysis is simplified greatly by assum-

ing linear profiles of pressure and concentrations across the

electrode. We have justified this assumption by solving the

two-dimensional dusty-gas model over a large range of

electrode structures and fuel-cell operating conditions. Once

the concentration gradients are evaluated, a new estimate for

the concentrations at the TPB can be found (assuming linear

concentration profiles and a known electrode thickness).

With the new estimated concentrations a new pressure

gradient can be found. This iteration converges very

rapidlytypically needing only two or three iterations.

Assuming a current density ie and a global electroche-

mical reaction, the molar flux of the gas species through the

anode and cathode can be evaluated as:

Ng

k;a nf0nf;k n

00f;k

ie

neF; (23)

Ngk;c no 0no;k n00o;k ie

neF: (24)

Note that the molar flux must be taken as zero for any inert

species that does not participate in the electrochemical

reactions (e.g. nitrogen in the cathode). Appendix A pro-

vides examples.

2.4. Activation polarization and the ButlerVolmer

equation

In the chemical reaction process, the reactants must usually

overcome an energy barrier, that is, the activation energy. In

charge-transfer electrochemical reactions, the reactants mustovercomenot only a thermal energybarrier but also an electric

potential. Thus, a portion of the potential cell voltage is lost

in driving the electrochemical reactions that transfer the

electrons to or from the electrodes. This activation polariza-

tion is a very important loss when the electrochemical reac-

tions are controlled by the slow electrode kinetics.

Assuming a single rate-controlling reaction, the Butler

Volmer equation [20] provides a relationship between the

current density and the charge-transfer overpotential as:

ie i0 exp aanBVe FZact

RT exp acnBVe FZact

RT : (25)This equation represents the net anodic and cathodic current

due to an electrochemical reaction. The activation over-

potential Zact is potential difference above the equilibrium

electric potential between the electrode and the electrolyte.

The factor nBVe is the number of electrons transferred in the

single elementary rate-limiting step that the ButlerVolmer

equation represents. It is quite common to assume nBVe 1.The anodic and cathodic asymmetric factors (also called

charge-transfer coefficients) aa and ac determine the relative

variations of the anodic and cathodic branches of the total

current with an applied overpotential Zact. The anodic and

cathodic charge-transfer coefficients aa

and ac

depend on the

electrocatalytic reaction mechanism and typically take

values between zero and one. Further, they are constrained

by aa ac 1. The exchange current density i0 is thecurrent density of the charge-transfer reaction at the

dynamic equilibrium electric potential difference Eeq. In

other words, at the equilibrium potential, the forward and

reverse current densities are equal at i0. A high exchange

current density implies the reaction proceeds rapidly upon

varying the potential from its equilibrium value. There is an

activation overpotential for the anode and the cathode.

Consequently, there are charge-transfer coefficients aa and

ac for both the anode and the cathode.



7/14

There are two potentially interesting limits of the Butler

Volmer equation. At very high surface activation overpo-

tential one of the exponential terms in ButlerVolmer equa-

tion becomes negligible. The resulting Tafel equation is a

straight line on a semi-logarithmic plot, lnie $ Zact. Alter-natively stated:

Zact RT

aanBVe F lnie lni0 for

aanBVe FZactRT

) 1;

(26)

or

Zact RT

acnBVe F lnie lni0

for acn

BVe FZactRT

) 1: (27)

In an SOFC, the Tafel equation can often be used to calculate

the activation overpotential at the cathode. The second

limiting case is realized at very lower activation polariza-tion. In this case, the following linear current overpotential

relationship applies:

Zact RT

aa acneF

ie

i0

: (28)

This equation can often be applied to calculate the activation

potential at the anode of the SOFC system. In our model

neither of these limiting cases is actually used to determine

the activation overpotentials. Rather, the ButlerVolmer

equation is solved iteratively in its native form.

The exchange current density i0 is a measure of the

electrocatalytic activity of the electrodeelectrolyte inter-face or TPB for a given electrochemical reaction. In fact,

the exchange current density is a crucially important factor

for determining the activation overpotential Zact. However,

i0 is not a simple constant parameter, but its value may

depend on the operating conditions and material properties,

including concentrations of reactants and products adjacent

to the electrode, temperature, pressure, the microstructure

and electrocatalytic activity of the electrode, and even

the conductivity of the electrolyte [21]. Particularly for

composite SOFC electrodes with the mixed ionic and

electronic conduction, such as a NiYSZ for the anode

and a LSMYSZ for the cathode, the electrochemical

reaction zone can be extended from the electrolyte-elec-

trode interface into the electrode. Thus, the charge-transfer

resistance can be reduced and the exchange current density

can be enhanced. Furthermore, the exchange current den-

sity for the composite electrode becomes a function of the

electrochemical properties of the electrolyte and electrode

and the effective TPB length. Thus, microstructural proper-

ties of the composite electrode, such as the grain size and

volume fraction of the electrolyte, will affect the exchange

current density. A steady-state currentvoltage experiment

can measure i0 at a particular operating condition. How-

ever, it is very difficult to obtain a general analytical

expression for i0, which requires detailed understanding

of the elementary thermal and electrochemical reaction

mechanisms occurring at the electrode and the microstruc-

ture of the electrode.

In our model we provide flexibility for the exchange

current density to depend on the reactant concentrations

as:

i0 i00

YKk1

Xgkk ; (29)

where i00 is a constant and gk and Xk the reaction order and

mole fraction (at the electrodeelectrolyte interface) for the

kth species, respectively. Although this formulation provides

flexibility in the model, it is still far from a general descrip-

tion of the elementary charge-transfer reactions steps.

Furthermore, it is difficult to determine the reaction orders

for any particular system.

At the cathode of an SOFC system, the gas-phase oxygen

reduction and ion incorporation into the electrolyte is a

complex process that involves a series of elementary reac-

tions steps, including the oxygen adsorption and dissociation

on the surface, diffusion of the intermediate species on

the surface, charge transfer, and oxygen incorporation into

the lattice of the electrolyte. Any of the reaction steps may

be the rate-limiting reaction for the overall process, depend-

ing on the operating conditions (temperature, oxygen partial

pressure), and material properties, and microstructure of

the electrode (catalytic activity, pore size, porosity, tortuos-

ity, and permeability). For the LSMYSZ composite cath-

ode, Kim et al. [22] have indicated that: (1) i0 $ p3=8O2

if

the rate-determining step is the charging step of adsorbedoxygen; (2) i0 $ p

1=4O2

if the rate-determining step is diffusion

of charged surface oxygen to the TPB; and (3) i0 does not

depend on pO2 if the rate-determining step is incorporation

of oxygen ions into the electrolyte. For the PtYSZ cathode,

the exchange current density i0 was shown to be proportional

to p1=4O2

at low pO2 and p1=4O2

at high pO2 [23]. Uchida et al.

[21] indicated that i0 may be proportional to the ionic

conductivity of the electrolyte for both the LSMYSZ

and PtYSZ cathodes depending on the operating tempera-

tures.

The situation is more complex at the anode of an SOFC

system. Reaction of a hydrocarbon fuel may involve direct

electrochemical oxidization [24], steam and/or dry reform-

ing, pyrolysis, and deposit formation. The general thermal

and electrochemical reaction steps may include dissocia-

tive adsorption on surfaces, surface and/or charge-transfer

reactions among the adsorbed intermediate species, surface

and/or bulk diffusion of the adsorbed species, and deso-

rption of products. Even for hydrogen, the simplest and

most widely studied fuel, there is still considerable debate

about the underlying electrochemical processes [5,7,25

27]. The rate-determining reaction step of the hydrogen

oxidation at the anode may be the dissociative adsorption

of hydrogen, the formation of hydroxyl, a charge-transfer



8/14

reaction, or the desorption of water, all of which can

depend on the operating conditions (such as the tempera-

ture, pressure, and species concentrations), macro- and

microstructures, and material properties of the electrode.

Additionally thermal and electrochemical reactions may

occur on the surface and in the bulk of the metal and/or

ceramic of the composite cermet electrode. Thus, there isnot a simple means to represent the relationship between

the exchange current density and the species concentra-

tions at the anode.

There are ways to measure the exchange current density

(or infer its value from related measurements). Based on the

ButlerVolmer equation, a charge-transfer resistance Rct can

be expressed as:

R1ct Atpb@ie@Z

Xk;T

i0AtpbnBVe F

RT

aa exp aanBVe FZ

RT

ac exp acnBVe FZ

RT

: (30)

At the zero activation overpotential:

Rct RT

i0Atpbaa acnBVe F; (31)

which presents a relationship between the exchange current

density i0 and the charge-transfer resistance Rct. The charge-

transfer resistance can be determined from EIS measure-

ments [6,13].

2.5. Ohmic polarization

Sources of ohmic losses in a fuel cell are the resistance to

the ion flow in the electrolyte and the resistance to the

electronic flow in the electrode. Due to typically high metal

loading resistive electric conductivity in the electrodes is

high, which limits ohmic losses in the electrodes. Thus,

ohmic polarization in an SOFC system is typically domi-

nated by ion resistance through the electrolyte. Such losses

can be reduced by decreasing the electrolyte thickness and

enhancing its ionic conductivity.

Since electric current flow in the electrolyte and electro-

des obey Ohms law, the ohmic losses can be expressed

simply as:

Zohm ieRtot; (32)

where Rtot is the total area-specific cell resistance, including

the area-specific resistances in the electrodeRed and the solid

electrolyte Rel. The ionic conductivity of the electrolyte can

usually expressed as:

sel s0T1 exp

Eel

RT

; (33)

where Eel is the activation energy for the ionic transport.

Thus, the specific ohmic resistance of the electrolyte with

thickness of Lel can be obtained as:

Rel Lel

sel: (34)

There are several opportunities for interface overpoten-

tials with in the MEA structure. For example, anywhere

two materials are physically bonded together there canbe a resistance the causes an interface overpotential,

Zinterface.

If the electrolyte has electrical conductivity, there is a

possibility for a leakage current that may be represented as

a leakage overpotential Zleakage. In our model:

Zleakage Z0leakage 1

i

imax

; (35)

where Z0leakage is the leakage overpotential at open circuit and

imax the maximum current density for the cell. Once the

Nernst potential is known, Z0leakage can be observed from

measurement of the open-circuit potential. The value of imaxcan be determined from the model by calculating current

density at which the cell power density vanishes.

3. Solution algorithm

The model is solved by determining the cell voltage for

a given current density, or vice versa. In general, the cell

potential Ecell is expressed as the difference between the

Nernst cell potential E and the sum of all the relevant

overpotentials, which depend on current density (Eq. (1)).

For a given cell configuration, with specified geometry and

material properties, as well as operating conditions (e.g.

pressure, temperature, and fuel and oxidizer composition),

each of the overpotentials must be evaluated as a function of

current density. Since the overpotential is usually an implicit

function of current density an iteration is needed. Most of the

iterations are accomplished using a NewtonRaphson

method. The solution procedure follows several steps as

follows:

Calculate the stoichiometric coefficients of the globalelectrochemical reaction by solving element balance

equations. This depends on the specific fuel and oxidizer

composition.

Based on the stoichiometric coefficients, evaluate thecharge transferred ne and the Nernst potential E.

Evaluate the species molar flux through the anode and thecathode (N

ga;k and N

gc;k). These are all evaluated explicitly

from the current density and the stoichiometry (Eqs. (23)

and (24)).

Calculate the species compositions at the electrodeelectrolyte interfaces ( Xk

s) from the DGM at given

species molar flux and species compositions in the chan-

nels. This is an iterative procedure.

Evaluate the concentration overpotentials for both elec-trodes (Zconc;a and Zconc;c) once the interface concentra-



9/14

tions are known (Eqs. (15) and (16)). This is an explicit

evaluation.

With a specified exchange current density i0 at the anodeand cathode, invert the ButlerVolmer equation to obtain

the activation overpotentials Zact;a and Zact;c. This is an

iterative procedure.

Calculate the ohmic overpotential Zohm in the electrolyte(Eq. (32)). This is an explicit procedure.

Determine the leakage overpotential Zleakage (Eq. (35)).Assuming that Z0leakage is specified, the value of imax must

be determined iteratively. The maximum current density

is achieved when the net voltage is zero. The value ofimaxdepends on the cell parameters as well as all the other

overpotentials. For a given current density ie, evaluate the

leakage overpotential.

With all the overpotentials in hand, evaluate the cellvoltage Ecell.

The procedure described above assumes that the cell

voltage is to be determined once a current density is

specified. To create a voltagecurrent plot, for example,

the procedure is repeated for a range of current densities.

When the MEA model is incorporated into a larger system-

level fuel-cell model, it is usually more natural to specify the

cell operating voltage and determine the local current den-

sity. In this case an outer iteration is implemented around the

whole procedure outlined above. Despite the nested itera-

tions, the convergence is very rapid.

4. Analysis of a methane-fueled SOFC

The objective of this section is to apply the model to a

particular anode-supported SOFC system operating with

methane as the fuel. Table 1 lists model parameters of the

unit cell. For an anode-supported SOFC system, both the

electrolyte and the cathode are much thinner than the anode.

Thus, it may be anticipated that the electrolyte, ohmic and

cathode-concentration overpotentials will be relatively

small. The transport parameters for calculating the ohmic

overpotentials in the electrolyte (Table 1) are based on the

oxygen-vacancy conductivity [28].

Assuming pure CH4 as a fuel and air as the oxidizer, Fig. 4

illustrates the predicted cell voltage and the power density as

a function of the current density. The peak power is seen

to be about 0:4 W/cm2 at about 0:4 V. Fig. 4 also showsthe contributions of all the overpotentials as shaded regions.

For this particular system, the cathode activation represents

the largest overpotential. It is followed by the anode-activa-

tion, ohmic, and anode-concentration overpotentials. The

cathode-concentration overpotential is so small that it does

not show on the plot. All the overpotentials increase with

increasing current density. The sum of the overpotentials

reaches the Nernst potential (in this case, about 1:15 V) atthe highest current density (ie % 2:25 A/cm

2) where the

power is reduced to zero. In other words, the cell ceases

to produce power when the overpotentials just equal the cell

potential.

4.0.1. Fuel utilization

Because fuel is consumed by electrochemical reaction as

it flows through the anode channel, the fuel stream is diluted

by product species (CO2 and H2O). Furthermore, as fuel is

Table 1

Parameters to represent a CH4 SOFC system

Parameter Value

Operating temperature (T, C) 750

Operating pressure (p, atm) 1

Anode

Thickness (La, mm) 1000

Porosity (fg) 0.35

Tortuosity (tg) 3.50

Average pore radius (rp, mm) 1

Average particle diameter (dp, mm) 10

Exchange current density (i0, A/cm2) 0.40

Charge-transfer coefficient (aa) 0.50

Number of electrons (nBVe ) 1

Cathode

Thickness (Lc, mm) 50

Porosity (fg) 0.35


Average pore radius (rp, mm) 1

Average particle diameter (dp, mm) 10

Exchange current density (i0, A/cm2

) 0.13Charge-transfer coefficient (aa) 0.50

Number of electrons (nBVe ) 1

Electrolyte

Thickness (Lel, mm) 20

Activation energy of O2 (Eel, J/mol) 8.0E4Pre-factor of O2 (s0, S/cm) 3.6E5Leakage overpotential (Zmax, V) 0.0

Fig. 4. Cell voltage, overpotentials, and power density as a function of the

current density for CH4 as a fuel in an SOFC with parameters stated in

Table 1.



10/14

used, the Nernst potential decreases (Fig. 3). The various

overpotentials are also affected by the fuel depletion. Fig. 5

illustrates the cell performance for situations where the fuel

is 50, 75, and 85% used. In this example, the exchange

current densities i0 at the anode and cathode are set to be

constants (Table 1). Furthermore, the cathode channel is

presumed to be pure air (i.e. no oxygen depletion). At 75 and

85% fuel utilization there are seen to be critical current

densities at which the power abruptly terminates. In these

cases (approximately 1:57 A/cm2 for the 75% case andapproximately 0:88 A/cm2 for the 85% case), the anode-

concentration overpotential becomes so large that no fuelcan reach the anode three-phase boundary. As a result the

cell cannot operate.

The cell represented by Fig. 5 would operate well at

very high fuel utilization. For example, if the operating

voltage was set at 0:5 V the current density would varyrelatively little (approximately between 0:6 a n d 0:8 A/cm2) with corresponding variations in power density

(approximately between 0:42 and 0:32 W/cm2) as thecomposition in the fuel channel varies between pure

methane and a highly diluted stream where the fuel is

85% depleted.

Fig. 6 presents the result of another simulation where all

parameters except one are the same as for the simulation

shown in Fig. 5. The cell represented by Fig. 6 presumes that

the anode exchange current density is first order in the fuel

concentration. That is:

i0 i00XCH4 ; (36)

where XCH4 is the fuel mole fraction and i00 0:65 A/cm

2.

As the fuel is depleted, the exchange current density drops

proportionally. In this case, the results are strikingly differ-

ent. As the fuel is depleted the current density and the power

density are significantly diminished. The anode-activation

overpotential dominates the losses. It would likely be

impractical to operate such a cell at a fuel utilization greater

than about 50%.

5. Interpreting experimental observations

Gorte et al. [1,24] have measured SOFC performance for a

variety of fuels using a YSZ electrolyte and a CuceriaYSZ

cermet anode. Fig. 7 reproduces some of the reported

measurements together with results from our model.

Table 2 lists parameters used in our MEA model to fit

the experimental data for the case of H2 as the fuel. Some ofthe physical and operating parameters are reported [1,24],

but we have estimated others. The parameters at the cathode

are chosen such that the concentration overpotential is

very small. The transport parameters for calculating the

Fig. 5. Cell voltages and power densities as a function of the current

density for fuel CH4 at a different percentage of fuel utilization. The

exchange current density is set to be constant.

Fig. 6. Cell voltages and power densities as a function of the current

density for CH4 at a different percentage of fuel utilization. The exchange

current density is assumed to be the first order in the fuel concentration.

Fig. 7. The MEA model is used to represent a range of experimental data

of currentvoltage and power densities for an SOFC from Gorte et al.

[1,24] by varying a few parameters.



11/14

ohmic overpotentials in the electrolyte are based on the

oxygen-vacancy conductivity of YSZ [28]. To match the

experimental data for cases of CH4 and C4H10, the para-meters: porosity, exchange current density, reaction orders

and the leakage overpotential are adjusted as indicated in

Table 3.

As illustrated in Fig. 7, the experimental measurements

can be represented very well by our model through varying a

few parameters. We must hasten to caution, however, that we

claim no uniqueness in the set of parameters used. Moreover,

although the model can match the data, it is difficult to

support some of the best-fit parameters on physical

grounds. Nevertheless, we believe that one can gain valuable

insights through interpreting experimental observations in

the context of a quantitative, physically based model.

As shown in Fig. 7, the measured open-circuit cell

potential for the H2 case is very close to the theoretical

Nernst potential. However, for the CH4 and C4H10 cases the

measured open-circuit potentials are approximately 0.2 V

lower than the corresponding theoretical Nernst potentials.

Therefore, we have assigned a leakage overpotential to

capture the measured open-circuit potential (Table 3).

Close inspection reveals that the currentvoltage curves

for these three fuels have substantially different functional

forms. The varying shapes infer that the overpotential lossesfor these three fuels are dominated by different physico-

chemical processes. For the H2 fuel, the nearly straight-line

shape (for the entire range of current densities), indicates

that the major polarization may be due to the ohmic loss at

the electrolyte rather than the activation overpotential and

concentration overpotential at the anode. For CH4, the data

shows a substantial upward curvature at low current density

but a nearly linear function at high current density. In this

case, the major overpotential loss may be caused by the

relatively low anode activation. In the case of C4H10, the

currentvoltage curve is again functionally different from

the other fuels, particularly at the high current density. The

relatively lower power density indicates that the activation

overpotential is very important, and the downward curvature

at the high current density indicates a significant anode-

concentration overpotential.

Since the concentration overpotentials appear to be rela-

tively small for cases of H2 and CH4, we have used the

measured values for porosity and pore size. In these cases,

we have varied the exchange current densities and reaction

orders to fit these measurements. However, for the case of

C4H10 fuel, the anode-concentration overpotential appears

to dominate the potential losses, particularly at the high

current density. Consequently, it appears that the gas-trans-

port resistance through the anode must be increased. Themost direct way to increase transport resistance is to reduce

the anode porosity. A very low porosity off % 2% is neededto reproduce the observed strong curvature in the data. Since

the measured porosity is f % 50%, one must seriouslyquestion the physical validity of the porosity needed to fit

the data. It is more likely that there may be some important

physical process that is not represented in the model as it

currently stands. For example, is it possible that there is

significant catalytic reaction within the anode structure? If

so, perhaps it is not the parent butane fuel that reacts

electrochemically, but rather some reaction products. Cer-

tainly if one of the essential underpinnings of the model (e.g.

direct electrochemical oxidation of the fuel) is incorrect,

then inferences from the data-fitting procedure must be

questioned.

Table 2

Parameters for matching the H2 experimental data by Gorte et al. [1,24]

Parameter Value

Operating temperature (T, C) 700

Operating pressure (p, atm) 1

Anode (CuceriaYSZ)

Thickness (La, mm) 400

Porosity (fg) 0.52


Average pore radius (rp, mm) 1.0

Average particle diameter (dp, mm) 3.0

Exchange current density (i00, A/cm2) 0.0055


Number of electrons (ne) 1

Cathode (LSMYSZ)

Thickness (Lc, mm) 50

Porosity (fg) 0.35


Average pore radius (rp, mm) 1.0

Average particle diameter (dp, mm) 3.0

Exchange current density (i00, A/cm

2

) 0.75Reaction order of O2 0.5


Number of electrons (ne) 1

Electrolyte (YSZ)

Thickness (Lel, mm) 60

Activation energy of O2 (Eel, J/mol) 8.0E4Pre-factor of O2 (s0, S/cm) 3.6E5Leakage overpotential (Zmax, V) 0.07

Table 3

Parameters adjusted to match the experiment data by Gorte et al. [1,24]

Fuel type Reaction order (gk) f i00 (A/cm

2) Z0leakage (V)

H2 CH4 C4H10 H2O CO2

Hydrogen (H2) 1.0 0.0 0.0 1:0 0.0 0.520 0.0055 0.07Methane (CH4) 0.0 1.0 0.0 0:5 0.0 0.520 0.0045 0.20Butane (C4H10 0.0 0.0 1.0 1:0 0.0 0.021 0.0090 0.23



12/14

6. Summary and conclusions

We have presented a physically based mathematical model

that represents a unit cell of a fuel-cell membrane-electrode

assembly. It is written to accommodate general descriptions

of fuel and oxidizer mixtures. It considers a variety of anode

and cathode overpotential losses that include activation,concentration, ohmic, and leakage. The activation overpo-

tentials are described in terms of the ButlerVolmer equation.

Concentration overpotentials in the porous electrode struc-

tures are described in terms of a dusty-gas model. Ohmic

losses may stem from ion resistance in the electrolyte or

electrical resistance at material interfaces.

The model can be exercised to produce voltagecurrent

relationships for a certain cell configurations and specified

operating conditions. These simulations are valuable in

helping to understand the competing physical processes that

are responsible for controlling cell performance. Such

understanding can assist in cell design and optimization

as well as interpreting experimental observations.

The objective of this paper is primarily to describe the

MEA model and the underlying assumptions. The model can

be used as a stand-alone model. However, the software is

written to be incorporated as a submodel into larger system-

level models that also consider the fluid flow and thermal

transport for a full fuel-cell system.

The model presented here assumes direct electrochemical

oxidation of the fuel species. Thus, while there is multi-

component transport of fuel, oxidizer, and product species

within the electrode structures, there is no homogeneous or

catalytic reaction among these species. Although there is

reported experimental evidence of DECO [24] for hydro-carbon fuels in certain SOFC systems, there is also reason to

anticipate that catalytic reaction could occur within porous

cermet electrode structures. Thus, in future work we intend

to extend the models to accommodate this possibility.

Acknowledgements

This work has been supported by DAPRA through the

Palm Power program and by the DoD Multidisciplinary

University Research Initiative (MURI) program adminis-

tered by the Office of Naval Research under Grant N00014-

02-1-0665. We gratefully acknowledge frequent and sub-

stantial collaboration with Prof. Greg Jackson (University of

Maryland) and Prof. David Goodwin (Caltech). We also

acknowledge close and beneficial collaborations with grad-

uate student Kevin Walters and colleagues at ITN Energy

Systems, especially Neal Sullivan and Bill Barker.

Appendix A

This paper develops and discusses a general formalism

beginning with a global electrochemical reaction for direct

electrochemical oxidation. To maintain generality there is a

certain level of abstract nomenclature that must be used. The

objective of this appendix is to assist understanding and

interpreting the nomenclature by working through specific

examples.

A.1. C4H10air SOFC

The overall electrochemical reaction is stated as:

C4H10 13

2 0:210:21O2 0:79N2

! 4CO2 5H2O 13 0:79

2 0:21N2; (A.1)

which combines the electrochemical reaction in the anode:

C4H10 13O2 ! 4CO2 5H2O 26e

; (A.2)

and the electrochemical reaction in the cathode:

132 O2 26e ! 13O2: (A.3)

The fuel and oxidizer compositions are specified by setting

the non-zero nf;k and no;k as:

nf;C4H10 1; (A.4)

no;O2 0:21; no;N2 0:79: (A.5)

Balancing the reaction yields the non-zero stoichiometric

coefficients as:

nf0 1; (A.6)

no0

13

2 0:21

; (A.7)

n00f;CO2 4; n00f;H2O

5; (A.8)

n00o;N2 13 0:79

2 0:21: (A.9)

The non-zero charge transfer per mole of each reactant

species are:

zf;CH4 26; zo;O2 4: (A.10)

With the reaction balanced and the charge transfers

assigned, the number of electrons transferred by the global

reaction follows from Eq. (4) as:

ne XKk1

nf0nf;kzf;k 26: (A.11)

The concentration overpotentials are expressed as:

Zconc;a RT

26Fln

C4H10

C4H10s

4 ln

CO2

CO2s

5 lnH2O

H2Os

; (A.12)

Zconc;c RT

4Fln

O2

O2s

: (A.13)



13/14

For the butaneair case, the species molar fluxes in the anode

are:

NgC4H10

ie

26F; N

gH2O

5ie

4F; N

gCO2

4ie

8F;

(A.14)

and in the cathode are:

NgO2

ie

4F; N

gN2

0: (A.15)

A.2. Fuel-mixtureair SOFC

This example considers the anode channel to have a

mixture of fuel (CH4), products (H2O and CO2), and oxygen

(O2). At some particular point in the fuel-cell system the

mole ratios in the anode channel are presumed to be

CH4:H2O:CO2:O21.0:0.3:0.15:0.05.This oxygen is explicitly added to the anode streamit is

not coming through the MEA. The oxidizer is presumed tobe air. The overall electrochemical reaction can be repre-

sented as follows:

CH4 0:3H2O 0:15CO2 0:05O2

20:21 0:21O2 0:79N2

! 2:3H2O 1:15CO2 0:05O2 2

0:210:79N2:

(A.16)

The half-cell reactions combine the electrochemical reaction

in the anode:

CH4 4O2 ! CO2 2H2O 8e

; (A.17)

and the electrochemical reaction in the cathode:

2O2 8e ! 4O2: (A.18)

The fuel and oxidizer compositions are specified by setting

the non-zero nf;k and no;k as:

nf;CH4 1; nf;H2O 0:3; nf;CO2 0:15;

nf;O2 0:05; (A.19)

no;O2 0:21; no;N2 0:79: (A.20)

Balancing the reaction yields the non-zero stoichiometric

coefficients as:

nf0 1; no

0 20:21 ; (A.21)

n00f;CO2 1:15; nf;H2 O00 2:3; n00f;O2 0:05;

n00o;N2 2 0:79

0:21: (A.22)

The non-zero charge transfer per mole of each reactant

species are:

zf;CH4 8; zo;O2 4: (A.23)

With the reaction balanced and the charge transfers

assigned, the number of electrons transferred by the global

reaction follows from Eq. (4) as:

ne XKk1

nf0nf;kzf;k 8: (A.24)

The concentration polarizations at the anode and cathode are

expressed as:

Zconc;a RT

8Fln

CH4

CH4s

ln

CO2

CO2s

2 lnH2O

H2Os

; (A.25)

Zconc;c RT

4Fln

O2

O2s

: (A.26)

The species molar fluxes in the anode are:

NgCH4

ie

8F; N

gH2O

ie

4F;

NgCO2 ie8F;NgO2 0; (A.27)

and in the cathode are:

NgO2

ie

4F; N

gN2

0: (A.28)

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a general mathematical model for analyzing the performance

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